ECO290E: Game Theory

1. ECO290E: Game Theory Lecture 12 Static Games of Incomplete Information

2. Review of Lecture 11 In the repeated Bertrand games, the following “trigger” strategies achieve collusion if δ≥1/2. • Each firm charges a monopoly price until someone undercuts the price, and after such deviation she will set a price equal to the marginal cost c, i.e., get into a price war.

3. Finite Repetitions Q: If the Bertrand games are played only finitely, i.e., ends in period T, then collusion can be sustained? A: NO! • In the last period (t=T), no firm has an incentive to collude since there is no future play. The only possible outcome is a stage game NE. • In the second to the last period (t=T-1), no firm has an incentive to collude since the future play will be a price war no matter how each firm plays in period T-1. • By backward induction, firms end up getting into price wars in every period. • No collusion is possible!

4. Finitely Repeated Games • If the stage game G has a unique NE, then for any T, the finitely repeated game G(T) has a unique SPNE: the NE of G is played in every stage irrespective of the history. • If the stage game G has multiple NE, then for any T, any sequence of those equilibrium profiles can be supported as the outcome of a SPNE. Moreover, non-NE strategy profiles can be sustained as a SPNE in this case.

5. Games of Incomplete Information • In a game of incomplete information, at least one player is uncertain about what other players know, i.e., some of the players possess private information, at the beginning of the game. • For example, a firm may not know the cost of the rival firm, a bidder does not usually know her competitors’ valuations in an auction. • Following Harsanyi (1967), we can translate any game of incomplete information into a Bayesian game in which a NE is naturally extended as a Bayesian Nash equilibrium.

6. Cournot Game with Unknown Cost • Firm 1’s marginal cost is constant (c), while firm 2’s marginal cost takes either high (h) with probability θ or low (l) with probability 1-θ. • Firm 1’s strategy is a quantity choice, but firm 2’s strategy is a complete action plan, i.e., she must specify her quantity choice in each possible marginal cost. • Assume each firm tries to maximize an expected profit.

7. Calculation • It is important to consider the types of player 2 as separate players. • Equilibrium strategies can be derived by the following maximization problems:

8. Solution • Notice that firm 2 will produce more (/less) than she would in the complete information case with high (/low) cost, since firm 1 does not take the best response to firm 2’s actual quantity but maximizes his expected profit.

9. Static Bayesian Games • Nature draws a type vector t, according to a prior probability distribution p(t). • Nature reveals i’s type to player i, but not to any other player. • The players simultaneously choose actions. • Payoffs are received.

10. Remarks • A belief about other players’ types is a conditional probability distribution of other players’ types given the player’s knowledge of her own type. • A (pure) strategy for a player is a complete action plan, which specifies her action for each of her possible type. • Bayes’ rule:

11. Bayesian Nash Equilibrium • A strategy profile s* is a Bayesian NE if which is equivalent to

12. Calculation • Maximizing RHS is identical to maximizing inside the brackets for all possible i’s type.